## Some Fun Stuff: Seinfeld and Optimal Stopping Times

Sometimes while you are proctoring exams, you realize that an episode of Seinfeld can be understood as an optimal stopping time problem and you write a short paper about it. Enjoy.

## Interest Rates and Investment

The conventional way of discussing monetary policy is by referencing the interest rate target of the central bank. This is also the way that monetary policy is communicated in the basic New Keynesian model. The idea is that the transmission of monetary policy is primarily through the interest rate. I would like to argue in this post that this is a problematic way of thinking about monetary policy and that the transmission mechanism of policy is unclear.

In the New Keynesian model, the real interest rate affects the time path of consumption through the consumption Euler equation. In particular, when the real interest rate falls, the household would want to save less and therefore would want to consume more. This increases real economic activity in the current period. If we add capital to the model, a lower interest rate encourages a greater investment in capital. Thus, if monetary policy can affect the real interest rate in the short run, then the interest rate target of the central bank can be used as a stabilization tool.

This investment mechanism, however, is questionable. It ignores how investment is actually done in the real world. We can illustrate this lesson with a simple example.

Suppose that there is a firm. The firm produces a product and is deciding whether to build a new factory to increase its production. Let V(t) denote the value of the factory at time t. The initial value of the project is $V(0) = V_0$. Now suppose that the value to the firm of building the factory is growing over time:

${{\dot{V}}\over{V}} = a$

It follows that the value of the factory at some arbitrary date in the future, say time T, is

$e^{aT} V_0$

Now suppose that the cost to build the factory is some fixed cost, $F$. The firm’s objective is to choose the optimal point in time to build the factory so as to maximize the expected discounted net value of the project:

$\max\limits_{T} e^{-rT} [e^{aT}V_0 - F]$

where $r > a$ is the real interest rate. The maximization problem implies that

$T^* = max\bigg[{{1}\over{a}} ln\bigg({{rI}\over{(r-a)V_0}}\bigg),0\bigg]$

Assuming that $T^* > 0$ (i.e. the optimal time to invest is not immediately), it is straightforward to see that when the real interest rate declines, it is beneficial to put off the investment further into the future.

We can understand the intuition behind this result as follows. In a standard model with capital, the marginal product of capital (net of some adjustment cost) is equal to the real interest rate. Thus, when the real interest rate falls, the firm wants to increase its investment in capital, but because it is costly to adjust that capital, it takes time for the capital stock to reach the firm’s desired level. In contrast, the framework presented above suggests that investment is an option and the firm has to decide when to exercise that option. In that case, a lower the real interest rate means that the future is more important (all else equal). But if the future is more important, then that increases the opportunity cost of exercising the option today. So the firm would want to wait to exercise the option.

So which way is best to think about interest rates and investment? The empirical evidence on the issue (albeit somewhat dated) seems to suggest that price variables, like the real interest rate, are not particularly useful in explaining investment (at least compared to other variables). So is this really the mechanism that should be emphasized in the conduct of monetary policy?

[I should note that this insight is (at least I thought) well known. This example is precisely the example provided by Dixit and Pindyck (1994). Countless other examples can be found in Stokey (2008).]

## Interest on Reserves and the Federal Funds Rate

The payment of interest on reserves is supposed to put a floor beneath the federal funds rate. Since banks can lend to one another overnight at the federal funds rate, they have a choice. The bank can either lend excess reserves to another bank at the federal funds rate or they can hold the reserves at the Federal Reserve and collect the interest the Fed pays on reserves. In theory, this means that the federal funds rate should never go below the interest rate on reserves. The reason is simple. No bank should have the incentive to lend at a lower rate than they would receive by not lending.

However, the effective federal funds rate has been consistently below the interest rate on reserves. How can this be so? Marvin Goodfriend explains:

The interest on reserves floor for the federal funds rate failed, and continues to fail to this day, because non-depository institutions (such as government-sponsored enterprises (GSEs) Fannie Mae and Freddie Mac, and Federal Home Loan Banks (FHLBs)) are authorized to hold overnight balances at the Fed, but are not eligible to receive interest on those balances. Hence, the GSEs and FHLSs [sp] have an incentive to try to earn interest on their overnight balances at the Fed by lending them to depositories eligible to receive interest on their reserve balances. The federal funds rate is thereby driven below interest on reserves to the point that depositories are willing to borrow from the GSEs and the FHLBs, deposit the proceeds at the Fed, and earn the spread between interest on reserves and the federal funds rate.

More here.

## What Does It Mean for the Natural Rate of Interest to Be Negative?

Talk of the zero lower bound has permeated the debate about monetary policy in recent years. In particular, there is one consistent story across a variety of different thinkers involving the difference between the natural rate of interest and the market rate of interest. Specifically, the argument holds that if the market rate of interest is higher than the natural rate of interest then monetary policy is too tight. With regards to the current state of the world, this is potentially problematic is the market rate of interest is zero, but needs to be lower.

I find this way of thinking about monetary policy to be quite odd for several reasons. First, conceivably when one talks about the natural rate of interest, the reference is to a real interest rate. New Keynesians, for example, clearly see the natural rate of interest as a real rate of interest (at least in their models). Second, the market rate of interest is a nominal rate. Thus, it is odd to say that the market rate of interest is above the natural rate of interest when one is nominal and one is real. I suppose that what they mean is that given the nominal interest rate and given the expectations of inflation, the implied real market rate is too high. But this seems to be an odd way to describe what is going on.

Regardless of this confusion, what advocates of this approach appear to be saying is this: when the market rate of interest is at the zero lower bound and the natural rate of interest is negative, unless inflation expectations rise, there is no way to equate the real market rate of interest with the natural rate.

But this brings me to the most important question that I have about this entire argument: Why is the natural rate of interest negative?

It is easy to imagine a real market interest rate being negative. If inflation expectations are positive and policymakers drive a nominal interest rate low enough, then the implied real interest rate is negative. It is NOT, however, easy to imagine the natural rate of interest being negative.

To simplify matters, let’s consider a world with zero inflation. The central bank uses an interest rate rule to set monetary policy. The nominal market rate is therefore equal to the real market interest rate. Thus, assuming that the central bank is pursuing a policy to maintain zero inflation, they are effectively setting the real rate of interest. Thus, the optimal policy is to set the interest rate equal to the natural interest rate. Also, since everyone knows the central bank will never create inflation, this makes the zero lower bound impenetrable (i.e. you cannot even use inflation expectations to lower the real rate when the nominal rate hits zero). I have therefore created a world in which a central bank is incapable of setting the market rate of interest equal to the natural rate of interest if the natural rate is negative. My question is, why in the world would we ever reach this point?

So let’s consider the determination of the natural rate of interest. I will define the natural rate of interest as the real rate of interest that would result with perfect markets, perfect information, and perfectly flexible prices (the New Keynesian would be proud, I think). To determine the equilibrium real interest rate, we need to understand saving behavior and we need to understand investment behavior. The equilibrium interest rate is then determined by the market in our perfect benchmark world. So let’s set up a really simple model of saving and investment.

Time is continuous and infinite. A representative household receives an endowment of income, y, and can either consume the income or save it. If they save it, they earn a real interest rate, r. The household generates utility via consumption. The household utility function is given as

$\int_0^{\infty} e^{-\rho t} u[c(t)] dt$

where $\rho$ is the rate of time preference and $c$ is consumption. The household’s asset holdings evolve according to:

$\dot{a} = y - c + ra$

where $a$ are the asset holdings of the individual. In a steady state equilibrium, it is straightforward to show that

$r = \rho$

The real interest rate is equal to the rate of time preference.

Now let’s consider the firm. Firms face an investment decision. Let’s suppose for simplicity that the firm produces bacon. We can then think of the firm as facing a duration problem. They purchase a pig at birth and they raise the pig. The firm then has to decide how long to wait until they slaughter the pig to make the bacon. Suppose that the duration of investment is given as $\theta$. The production of bacon is given by the production function:

$b = f(\theta)$

where f’,-f”>0 and b is the quantity of bacon produced. The purchase of the pig requires some initial outlay of investment, $i$, which is assumed to be exogenously fixed in real terms and then it just grows until it is slaughtered. The value of the pig over the duration of the investment is given as

$p = \int_{-\theta}^0 e^{-rt} i dt$

Integration of this expression yields

${{p}\over{i}} = {{1}\over{r}}(e^{r\theta} - 1)$

Let’s normalize the amount of investment done to 1. Thus, we can write the firm’s profit equation as

$\textrm{Profit} = f(\theta) - e^{r\theta}$

The firm’s profit-maximizing decision is therefore given as

$f'(\theta) = re^{r\theta}$

Given that the firm makes zero economic profits, it is straightforward to show that

$r = {{f'(\theta)}\over{f(\theta)}}$

So let’s summarize what we have. We have an inverse supply of saving curve that is given as

$r = \rho$

Thus, the saving curve is a horizontal line at the rate of time preference.

The inverse investment demand curve is given as

$r = {{f'(\theta)}\over{f(\theta)}}$

The intersection of these two curves determine the equilibrium real interest rate and the equilibrium duration of investment. Since the supply curve is horizontal, the real interest rate is always equal to the rate of time preference. So this brings me back to my question: How can we explain why the natural rate of interest would be negative?

You might look at the equilibrium conditions and think “sure the natural rate of interest can be negative, we just have to assume that the rate of time preference is negative.” While, this might mathematically be true, it would seem to imply that people value the future more than the present. Does anybody believe that to be true? Are we really to believe that the the zero lower bound is a problem because the general public’s preferences change such that they suddenly value the future more than the present?

But suppose you are willing to believe this. Suppose you think it is perfectly reasonable to assume that people woke up sometime during the recession and their rate of time preference was negative. There are two sides to the market. So what would happen to the duration of investment if the real interest rate was negative? From our inverse investment demand curve, we see that the real interest rate is equal to the ratio of the marginal product of duration over total production. We have made the standard assumption that the marginal product is positive, so this would seem to rule out any equilibrium in which the real interest rate was negative. But suppose at a sufficiently long duration, the marginal product is negative. We could always write down a production function with this characteristic, but how generalizable would this production function be? And why would a firm choose this actually duration when they could have chosen a shorter duration and had the same level of production?

Thus, the only way that one can believe that the natural rate of interest is negative is if they believe that individuals suddenly value the future more than the present and that in a perfect, frictionless world firms would prefer to undertake dynamically inefficient investment projects. And not only that, advocates of this viewpoint also think that the problem with policy is that we cannot use our policy tools to get us to a point consistent with these conditions!

Finally, you might argue that I have simply cherry-picked a model that fits my conclusion. But the model I have presented here is just Hirshleifer’s attempt to model the theories of Bohm-Bawerk and Wicksell, the economists that came up with the idea of the natural rate of interest. So this would seem to be a good starting point for analysis.

P.S. If you are interested in evaluating monetary policy within a framework like this, you should check out one of my working papers, written with Alex Salter.

## More on Germany

Tony Yates has a new post speculating on Germany’s attitude toward fiscal policy. Tony’s assumption is that the German’s opposition to a deal with Greece is rooted in their beliefs about the macroeconomy. In particular, the New Keynesian consensus never really took hold in Germany and therefore they don’t tend to look favorably on stabilization policies. Some might view Tony’s post as a condemnation of Germany, but I don’t view it as such. If you’re a New Keynesian, you probably think this is a bad thing. If you’re not a New Keynesian, or if you’re otherwise opposed to stabilization policy, you probably think this is a good thing. But regardless of whether it is good or bad, this is a possible explanation. Nonetheless, I don’t find this explanation all that convincing.

As I wrote in my previous post on the topic, I think that macroeconomists tend to look at the negotiations between Germany (and the EU more broadly) and Greece all wrong. Naturally, as macroeconomists, we tend to think about these negotiations in terms of the “big picture”. In other words, when macroeconomists look at the negotiations, they often think about the macroeconomic effects of a financial market collapse in Greece and the possible damage that results from various interlinkages. Those who see these costs as being very large then tend to advocate the importance of coming to an agreement. In addition, those who advocate stabilization policies think that such an agreement should also allow Greece to defer the costs until they have a chance to improve their economy through (you guessed it) stabilization policies.

I think that this is the wrong way to think about the negotiations. The negotiations have to be viewed in the context of game theory. The Germans and the Greeks are playing a dynamic game. Thus, Germany has an incentive to make sure that any deal that is reached between the EU and Greece is one that prevents these types of negotiations from happening in the future. In other words, Germany is trying to minimize the costs associated from moral hazard in the future. This isn’t just about Greece, this is about setting a precedent for all future negotiations. When you think about the negotiations in this context, I think that you come up with a better understanding of Germany’s behavior.

But I also think that once you think about the negotiations in terms of game theory, the supposed German opposition to fiscal stabilization doesn’t hold up very well as an explanation of the German response. For example, suppose that the Germans do believe in fiscal stabilization policies. Wouldn’t it then make sense to use austerity as a punishment mechanism for the bailout? If Germany’s true desire is to prevent such negotiations from taking place in the future, then they have an incentive to enact some sort of punishment on Greece in any deal that they reach. Imposing austerity would be one possible punishment. Even if the Germans don’t believe in fiscal stabilization, they know that the Greeks do. As a result, this threat is still a credible way of imposing costs in the game theoretic context because the expected costs to Greece are conditional on Greece’s expectations.

In short, whether or not Germany believes in economic stabilization policy, they are likely to pursue the same strategy within a game theoretic context. Thus, viewing this strategy on the part of the Germans doesn’t necessarily tell us anything about German beliefs.

## Resolving the Glasner-Sumner Dispute

David Glasner and Scott Sumner are arguing about whether saving = investment is an identity or an equilibrium condition. So I thought I would step in and resolve this dispute. Instead of using textbook accounting identities, let’s consider a framework everyone is familiar with — a two-period consumption model.

1. Consider a Robinson Crusoe economy. There is one guy on an island with production opportunities, but no market opportunities. For simplicity, think of a two-period model. In the first period, the individual receives an endowment, Y. The individual can invest that endowment to generate future production or consume the endowment. The individual transforms Y into P1, production now, and P2, production later. It follows that investment is defined as I = Y – P1. Savings is defined as S = Y – C1, where C1 is consumption in the first period. Since there is only one guy on the island, it must be true that P1 = C1. These decisions are both determined by the individual’s rate of time preference. Thus, S = I is an identity.

2. Consider the same guy on an island, but who now has market opportunities. Now we have the same definitions for saving and investment. Saving is

S = Y – C1
I = Y – P1

Note that with exchange opportunities, it is very unlikely that C1 = P1. Thus, at the individual level, savings probably doesn’t equal investment. Combining these conditions, we get

S = I + P1 – C1

for the individual. Now sum across all terms and we get

$\sum S = \sum I + \sum P1 - \sum C1$

Now in equilibrium, market-clearing requires that total production equals total consumption. Thus, market clearing implies that total savings is equal to total investment:

$\sum S = \sum I$

Saving = Investment is therefore an equilibrium condition.

3. Finally, David’s issue is that he doesn’t think that gross domestic income and gross domestic expenditure are the same thing. Empirically, he’s correct. This is why we have GDP Plus.

## Germany, Greece, and Rent-Seeking

Greece is currently seeking a bailout from the European Union. However, negotiations (at least as I write this) are at a standstill. Greece wants a bailout, but the new Greek government has indicated that it is unwilling to enact so-called austerity reforms. During the negotiations, the European Central Bank has given emergency funding to the Greek financial system. However, this funding is conditional on the negotiations between Greece and the EU. The finance ministers of various EU countries want Greece to commit to reducing their debt in line with their 2012 commitments. Since Greece appears unwilling to meet those requirements, the support from the ECB is likely to stop by the end of the month. Thus, Greece could face a significant financial crisis if no deal is reached.

In the midst of these negotiations, some have argued that the EU, and Germany specifically, do not appear to understand the magnitude of the situation. Paul Krugman, for instance, writes

As long as it stays on the euro, then, Greece needs the good will of the central bank, which may, in turn, depend on the attitude of Germany and other creditor nations.

But think about how that plays into debt negotiations. Is Germany really prepared, in effect, to say to a fellow European democracy “Pay up or we’ll destroy your banking system?”

[…]

Doing the right thing would, however, require that other Europeans, Germans in particular, abandon self-serving myths and stop substituting moralizing for analysis.

This last statement is particular telling. Many commentators agree with Krugman and view the Germans and other members of the EU as moralizing. In other words, they are not using economic analysis, but rather relying on their own views about right and wrong and how a government should operate. However, I would submit that rather than assuming that the Germans and other members of the EU are vindictive moralizers, an understanding of economics can actually teach us why EU members have taken their current position. But to understand why, we need to know something about rent-seeking.

Suppose that there are two countries, Germany and Greece. In addition, suppose that each of these countries have an endowment of resources, $R_i$, where $i=1$ will refer to Germany and $i=2$ will refer to Greece. Now let’s assume that each country can devote some amount of resources to production, $P_i$ and some amount of time to fighting with each other, $F_i$. It follows that each country has a resource constraint:

$R_i = P_i + F_i$

Now, let’s assume that the total production between the two countries is given as

$Y = (P_1^{1/s} + P_2^{1/s})^s$

where $s \geq 1$ is a measure of complementarity in production. One way to think about $s$ is that the higher its value, the most closely linked the two countries are in terms of international trade, production, etc.

Thus, we see that the countries can commit their resources to production or to fighting. The more each country contributes to production, the higher the total level of production. However, fighting with one another can also provide benefits (this obviously doesn’t have to refer to actual fighting, it could refer to negotiations like those that are ongoing). However, whereas increased production will cause an increase in the amount of production/income that is generated, fighting will only have an effect on the distribution of income.

Thus, each country faces a trade-off. The more resources they commit to production, the higher the level of income that is generated. The more resources they commit to fighting, the greater the distribution of the existing income they receive (but there is less income as a result). Thus each country has to choose the share of resources that they want to commit to production and fighting.

We will assume that there is a contest success function (as in Tullock, 1980) that is a function of the amount of resources that each country commits to fighting. For Germany, we assume that the contest success function is given as

$\mu_1 = {{F_1^m}\over{F_1^m + F_2^m}}$

where $m$ is an index of the decisiveness of the conflict. Correspondingly, for Greece $\mu_2 = 1 - \mu_1$.

Given these definitions, we can then define the distribution of income:

$Y_1 = \mu_1 Y$
$Y_2 = \mu_2 Y$

Now let’s assume that each country wants to maximize their own income $Y_i$, taking what the other country is doing as given. Thus, each country wants to maximize the following

$\max\limits_{P_i, F_i} {{F_i^m}\over{F_i^m + F_j^m}} [(P_1^{1/s} + P_2^{1/s})^s]$
$\textrm{s.t.} \hspace{2mm} R_i = P_i + F_i$

In equilibrium, it follows that

${{F_2 P_2^{(1-s)/s}}\over{F_1^m}} = {{F_1 P_1^{(1-s)/s}}\over{F_2^m}}$

Now let’s use this equilibrium condition to understand the interaction between Germany and Greece. Suppose that we simply choose resource endowments of $R_1 = 100$ and $R_2 = 50$ thereby assuming that Germany has twice as many resources as Greece. Now let’s consider the implications under two scenarios. First, we will consider the scenario in which $s = 1$. In this case, total production is just the sum of German and Greek production. It follows from equilibrium that

$F_1 = F_2$

Thus, in this case, Germany and Greece devote the same amount of resources to the fighting. However, since Germany has twice as many resources as Greece, it follows that Greece is devoting a larger percentage of their resources to fighting. In devoting resources to fighting, the two countries produce less total production. To see this, consider that if each country devoted all of their resources to production, total production would be 100 + 50 = 150. If we assume that $m = 1$, then $F_1 = F_2 = 37.5$. Thus, total production is 62.5 + 12.5 = 75. And yet this is their optimal choice given what they expect the other country to do!

As a result of the percentage of resources devoted to fighting, Greece gets 1/2 of the resulting production, despite only having 1/3 of the total resources. It should therefore be straightforward to understand why Greece is devoting so many resources to getting a larger bailout without bearing the costs of so-called austerity measures. In addition, it is important to note that it is in Germany’s best interest to devote resources to fighting, given what they expect Greece to do.

It is straightforward to show two other results with this framework. First, as $s$ increases, the weaker side has less of an incentive to fight whereas the stronger side has a greater incentive to fight. In other words, when production becomes more complementary, then the poorer side has less of an incentive to fight because of the linkages in production between their production effort and the other country. While the richer country has an incentive to fight more, the total resources devoted to fighting will fall.

Again, this informs the discussion about Germany and Greece in comparison to other countries. If you want to understand why there seems to be a greater conflict between Germany and Greece than between Germany and other EU countries, consider that Greece is Germany’s 40th largest trading partner, just after Malaysia and just ahead of Slovenia. All else equal the model described above suggests that we should expect more resources devoted to conflict.

The second characteristic has to do with the decisiveness of conflict. As $m$ increases, the conflict between the two countries can be considered more decisive. As the conflict becomes more decisive, the model predicts that the sides will have more of an incentive to devote to fighting. Thus, if Germany believes that this is (or should be) the last round of negotiations with Greece, then we would expect them to devote more resources to fighting.

So what is the point of this exercise?

The entire point of this exercise is to understand that Germany is, in fact, acting in their own economic interest given what they expect Greece to do. If we want to understand the positions of Germany and Greece, we need to understand strategic behavior. Germany’s refusal to simply give in to Greece’s demands and “do what’s best for Europe” ignores a lot of the aspects of what is going on here. Those who think that Germany is being too harsh ignore some of the key aspects of the framework discussed above.

First, one should note that the distribution of income in the framework above is always more equal to the distribution of resources. Thus, it pays for Greece to fight. However, Germany knows that it pays for Greece to fight and therefore Germany’s best strategic decision is to devote resources to fighting as well.

Second, by claiming that Germany is failing to do what is in the “best interests of Europe”, the critics are presuming that they know the social welfare function that needs to be maximized. Perhaps they are correct. Perhaps they are not. But even if these critics are correct, this doesn’t mean that Germany’s behavior is incomprehensible or that it is based on something other than economics. Rather the confusion is on the part of the critics who fail to understand that a Nash equilibrium may or may not be the socially desirable equilibrium. Germany’s rhetoric might be moralizing, but we can understand their behavior through an understanding of economics. And this is true whether you like Germany’s behavior or not.

[This post is an application of Hirshleifer’s Paradox of Power model. See here.]